SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5747-5781, October 2024. Abstract. We are interested in the gradient flow of a general first order convex functional with respect to the [math]-topology. By means of an implicit minimization scheme, we show existence of a global limit solution, which satisfies an energy-dissipation estimate, and solves a nonlinear and nonlocal gradient flow equation, under the assumption of strong convexity of the energy. Under a monotonicity assumption we can also prove uniqueness of the limit solution, even though this remains an open question in full generality. We also consider a geometric evolution corresponding to the [math]-gradient flow of the anisotropic perimeter. When the initial set is convex, we show that the limit solution is monotone for the inclusion, convex, and unique until it reaches the Cheeger set of the initial datum. Eventually, we show with some examples that uniqueness cannot be expected, in general, in the geometric case.
{"title":"[math]-Gradient Flow of Convex Functionals","authors":"Antonin Chambolle, Matteo Novaga","doi":"10.1137/22m1527556","DOIUrl":"https://doi.org/10.1137/22m1527556","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5747-5781, October 2024. <br/> Abstract. We are interested in the gradient flow of a general first order convex functional with respect to the [math]-topology. By means of an implicit minimization scheme, we show existence of a global limit solution, which satisfies an energy-dissipation estimate, and solves a nonlinear and nonlocal gradient flow equation, under the assumption of strong convexity of the energy. Under a monotonicity assumption we can also prove uniqueness of the limit solution, even though this remains an open question in full generality. We also consider a geometric evolution corresponding to the [math]-gradient flow of the anisotropic perimeter. When the initial set is convex, we show that the limit solution is monotone for the inclusion, convex, and unique until it reaches the Cheeger set of the initial datum. Eventually, we show with some examples that uniqueness cannot be expected, in general, in the geometric case.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5807-5839, October 2024. Abstract. This paper is concerned with geometric motion of a closed surface whose velocity depends on a nonlocal quantity of the enclosed region. Using the level set formulation, we study a class of nonlocal Hamilton–Jacobi equations and establish a control-based representation formula for solutions. We also apply the formula to discuss the fattening phenomenon and large-time asymptotics of the solutions.
{"title":"A Representation Formula for Viscosity Solutions of Nonlocal Hamilton–Jacobi Equations and Applications","authors":"Takashi Kagaya, Qing Liu, Hiroyoshi Mitake","doi":"10.1137/23m1608136","DOIUrl":"https://doi.org/10.1137/23m1608136","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5807-5839, October 2024. <br/> Abstract. This paper is concerned with geometric motion of a closed surface whose velocity depends on a nonlocal quantity of the enclosed region. Using the level set formulation, we study a class of nonlocal Hamilton–Jacobi equations and establish a control-based representation formula for solutions. We also apply the formula to discuss the fattening phenomenon and large-time asymptotics of the solutions.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5678-5722, August 2024. Abstract. We study an inverse problem of determining a time-dependent damping coefficient and potential appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of conformally transversally anisotropic manifolds, or in other words, compact Riemannian manifolds with boundary conformally embedded in a product of the Euclidean line and a transversal manifold. With an additional assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove that the knowledge of a certain partial Cauchy data set determines the time-dependent damping coefficient and potential uniquely.
{"title":"Partial Data Inverse Problem for Hyperbolic Equation with Time-Dependent Damping Coefficient and Potential","authors":"Boya Liu, Teemu Saksala, Lili Yan","doi":"10.1137/23m1588676","DOIUrl":"https://doi.org/10.1137/23m1588676","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5678-5722, August 2024. <br/> Abstract. We study an inverse problem of determining a time-dependent damping coefficient and potential appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of conformally transversally anisotropic manifolds, or in other words, compact Riemannian manifolds with boundary conformally embedded in a product of the Euclidean line and a transversal manifold. With an additional assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove that the knowledge of a certain partial Cauchy data set determines the time-dependent damping coefficient and potential uniquely.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5634-5677, August 2024. Abstract. We introduce a variant of the Smoluchowski coagulation equation as a kinetic equation with both position and velocity variables, which arises as the scaling limit of a system of second-order microscopic coagulating particles. We focus on the rigorous study of the [math] system in the spatially homogeneous case, proving existence and uniqueness under different initial conditions in suitable weighted spaces, investigating also the regularity of such solutions.
{"title":"Smoluchowski Coagulation Equation with Velocity Dependence","authors":"Franco Flandoli, Ruojun Huang, Andrea Papini","doi":"10.1137/22m1540594","DOIUrl":"https://doi.org/10.1137/22m1540594","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5634-5677, August 2024. <br/> Abstract. We introduce a variant of the Smoluchowski coagulation equation as a kinetic equation with both position and velocity variables, which arises as the scaling limit of a system of second-order microscopic coagulating particles. We focus on the rigorous study of the [math] system in the spatially homogeneous case, proving existence and uniqueness under different initial conditions in suitable weighted spaces, investigating also the regularity of such solutions.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5723-5745, August 2024. Abstract. We obtain an upper bound on the lowest magnetic Neumann eigenvalue of a bounded, convex, smooth, planar domain with moderate intensity of the homogeneous magnetic field. This bound is given as a product of a purely geometric factor expressed in terms of the torsion function and of the lowest magnetic Neumann eigenvalue of the disk having the same maximal value of the torsion function as the domain. The bound is sharp in the sense that equality is attained for disks. Furthermore, we derive from our upper bound that the lowest magnetic Neumann eigenvalue with the homogeneous magnetic field is maximized by the disk among all ellipses of fixed area provided that the intensity of the magnetic field does not exceed an explicit constant dependent only on the fixed area.
{"title":"A Geometric Bound on the Lowest Magnetic Neumann Eigenvalue via the Torsion Function","authors":"Ayman Kachmar, Vladimir Lotoreichik","doi":"10.1137/23m1624658","DOIUrl":"https://doi.org/10.1137/23m1624658","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5723-5745, August 2024. <br/> Abstract. We obtain an upper bound on the lowest magnetic Neumann eigenvalue of a bounded, convex, smooth, planar domain with moderate intensity of the homogeneous magnetic field. This bound is given as a product of a purely geometric factor expressed in terms of the torsion function and of the lowest magnetic Neumann eigenvalue of the disk having the same maximal value of the torsion function as the domain. The bound is sharp in the sense that equality is attained for disks. Furthermore, we derive from our upper bound that the lowest magnetic Neumann eigenvalue with the homogeneous magnetic field is maximized by the disk among all ellipses of fixed area provided that the intensity of the magnetic field does not exceed an explicit constant dependent only on the fixed area.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Simão Correia, Filipe Oliveira, Jorge Drumond Silva
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5604-5633, August 2024. Abstract. We consider the problem of establishing nonlinear smoothing as a general feature of nonlinear dispersive equations, i.e., the improved regularity of the integral term in Duhamel’s formula, with respect to the initial data and the corresponding regularity of the linear evolution, and how this property relates to local well-posedness. In a first step, we show how the problem generally reduces to the derivation of specific frequency-restricted estimates, which are multiplier estimates in the spatial frequency alone. Then, using a precise methodology, we prove these estimates for the specific cases of the modified Zakharov–Kuznetsov equation, the cubic and quintic nonlinear Schrödinger equation, and the quartic Korteweg–de Vries equation.
{"title":"Sharp Local Well-Posedness and Nonlinear Smoothing for Dispersive Equations through Frequency-Restricted Estimates","authors":"Simão Correia, Filipe Oliveira, Jorge Drumond Silva","doi":"10.1137/23m156923x","DOIUrl":"https://doi.org/10.1137/23m156923x","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5604-5633, August 2024. <br/> Abstract. We consider the problem of establishing nonlinear smoothing as a general feature of nonlinear dispersive equations, i.e., the improved regularity of the integral term in Duhamel’s formula, with respect to the initial data and the corresponding regularity of the linear evolution, and how this property relates to local well-posedness. In a first step, we show how the problem generally reduces to the derivation of specific frequency-restricted estimates, which are multiplier estimates in the spatial frequency alone. Then, using a precise methodology, we prove these estimates for the specific cases of the modified Zakharov–Kuznetsov equation, the cubic and quintic nonlinear Schrödinger equation, and the quartic Korteweg–de Vries equation.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5583-5603, August 2024. Abstract. We provide a rigorous validation that the infinite Calogero–Moser lattice can be well-approximated by solutions of the Benjamin–Ono equation in a long-wave limit.
{"title":"Approximation of Calogero–Moser Lattices by Benjamin–Ono Equations","authors":"J. Douglas Wright","doi":"10.1137/24m1629869","DOIUrl":"https://doi.org/10.1137/24m1629869","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5583-5603, August 2024. <br/> Abstract. We provide a rigorous validation that the infinite Calogero–Moser lattice can be well-approximated by solutions of the Benjamin–Ono equation in a long-wave limit.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5521-5582, August 2024. Abstract. This paper concerns the asymmetric transport observed along interfaces separating two-dimensional bulk topological insulators modeled by (continuous) differential Hamiltonians and how such asymmetry persists after numerical discretization. We first demonstrate that a relevant edge current observable is quantized and robust to perturbations for a large class of elliptic Hamiltonians. We then establish a bulk edge correspondence stating that the observable equals an integer-valued bulk difference invariant depending solely on the bulk phases. We next show how to extend such results to periodized Hamiltonians amenable to standard numerical discretizations. A form of no-go theorem implies that the asymmetric transport of periodized Hamiltonians necessarily vanishes. We introduce a filtered version of the edge current observable and show that it is approximately stable against perturbations and converges to its quantized limit as the size of the computational domain increases. To illustrate the theoretical results, we finally present numerical simulations that approximate the infinite domain edge current with high accuracy and show that it is approximately quantized even in the presence of perturbations.
{"title":"Approximations of Interface Topological Invariants","authors":"Solomon Quinn, Guillaume Bal","doi":"10.1137/23m1568387","DOIUrl":"https://doi.org/10.1137/23m1568387","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5521-5582, August 2024. <br/> Abstract. This paper concerns the asymmetric transport observed along interfaces separating two-dimensional bulk topological insulators modeled by (continuous) differential Hamiltonians and how such asymmetry persists after numerical discretization. We first demonstrate that a relevant edge current observable is quantized and robust to perturbations for a large class of elliptic Hamiltonians. We then establish a bulk edge correspondence stating that the observable equals an integer-valued bulk difference invariant depending solely on the bulk phases. We next show how to extend such results to periodized Hamiltonians amenable to standard numerical discretizations. A form of no-go theorem implies that the asymmetric transport of periodized Hamiltonians necessarily vanishes. We introduce a filtered version of the edge current observable and show that it is approximately stable against perturbations and converges to its quantized limit as the size of the computational domain increases. To illustrate the theoretical results, we finally present numerical simulations that approximate the infinite domain edge current with high accuracy and show that it is approximately quantized even in the presence of perturbations.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969442","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5474-5520, August 2024. Abstract. We are concerned with two-dimensional steady supersonic flows past Lipschitz wedges for the relativistic Euler equations. If the vertex angle of the upstream flow is less than the critical angle, determined by shock polar, then a shock wave is generated from the wedge vertex. When the total variations of the tangent angle of the boundary and the upstream flow are both suitably small, we establish global stability of entropy solutions, including a large 1-shock wave. Moreover, we obtain global nonrelativistic limits of the entropy solutions, and also investigate the asymptotic behavior of these solutions as [math]. It is worth mentioning that we demonstrate the basic properties of nonlinear waves for the two-dimensional steady relativistic Euler system, especially the geometric structure of shock polar.
{"title":"Steady Supersonic Flows Past Lipschitz Wedges for Two-Dimensional Relativistic Euler Equations","authors":"Min Ding, Yachun Li","doi":"10.1137/23m1600530","DOIUrl":"https://doi.org/10.1137/23m1600530","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5474-5520, August 2024. <br/> Abstract. We are concerned with two-dimensional steady supersonic flows past Lipschitz wedges for the relativistic Euler equations. If the vertex angle of the upstream flow is less than the critical angle, determined by shock polar, then a shock wave is generated from the wedge vertex. When the total variations of the tangent angle of the boundary and the upstream flow are both suitably small, we establish global stability of entropy solutions, including a large 1-shock wave. Moreover, we obtain global nonrelativistic limits of the entropy solutions, and also investigate the asymptotic behavior of these solutions as [math]. It is worth mentioning that we demonstrate the basic properties of nonlinear waves for the two-dimensional steady relativistic Euler system, especially the geometric structure of shock polar.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141887200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5445-5473, August 2024. Abstract. We extend the result of Kowalczyk, Martel, and Muñoz [J. Eur. Math. Soc. (JEMS), 24 (2022), pp. 2133–2167] on the existence, in the context of spatially even solutions, of asymptotic stability on a center hypersurface at the soliton of the defocusing power nonlinear Klein–Gordon equation with [math], to the case [math]. The result is attained performing new and refined estimates that allow us to close the argument for power law in the range [math].
{"title":"On Asymptotic Stability on a Center Hypersurface at the Soliton for Even Solutions of the Nonlinear Klein–Gordon Equation When [math]","authors":"Scipio Cuccagna, Masaya Maeda, Federico Murgante, Stefano Scrobogna","doi":"10.1137/23m1590871","DOIUrl":"https://doi.org/10.1137/23m1590871","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5445-5473, August 2024. <br/> Abstract. We extend the result of Kowalczyk, Martel, and Muñoz [J. Eur. Math. Soc. (JEMS), 24 (2022), pp. 2133–2167] on the existence, in the context of spatially even solutions, of asymptotic stability on a center hypersurface at the soliton of the defocusing power nonlinear Klein–Gordon equation with [math], to the case [math]. The result is attained performing new and refined estimates that allow us to close the argument for power law in the range [math].","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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